Whakaoti i te 2x(91+2-2x)=1080

$$2x(91+2-2x)=1080$$

Whakaoti mō x

$$x = \frac{3 \sqrt{481} + 93}{4} \approx 39.69878415$$

$$x = \frac{93 - 3 \sqrt{481}}{4} \approx 6.80121585$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tāpirihia te 91 ki te 2, ka 93.

$$2x\left(93-2x\right)=1080$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te 93-2x.

$$186x-4x^{2}=1080$$

Tangohia te 1080 mai i ngā taha e rua.

$$186x-4x^{2}-1080=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-4x^{2}+186x-1080=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, 186 mō b, me -1080 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-186±\sqrt{186^{2}-4\left(-4\right)\left(-1080\right)}}{2\left(-4\right)}$$

Pūrua 186.

$$x=\frac{-186±\sqrt{34596-4\left(-4\right)\left(-1080\right)}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$x=\frac{-186±\sqrt{34596+16\left(-1080\right)}}{2\left(-4\right)}$$

Whakareatia 16 ki te -1080.

$$x=\frac{-186±\sqrt{34596-17280}}{2\left(-4\right)}$$

Tāpiri 34596 ki te -17280.

$$x=\frac{-186±\sqrt{17316}}{2\left(-4\right)}$$

Tuhia te pūtakerua o te 17316.

$$x=\frac{-186±6\sqrt{481}}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$x=\frac{-186±6\sqrt{481}}{-8}$$

Nā, me whakaoti te whārite x=\frac{-186±6\sqrt{481}}{-8} ina he tāpiri te ±. Tāpiri -186 ki te 6\sqrt{481}.

$$x=\frac{6\sqrt{481}-186}{-8}$$

Whakawehe -186+6\sqrt{481} ki te -8.

$$x=\frac{93-3\sqrt{481}}{4}$$

Nā, me whakaoti te whārite x=\frac{-186±6\sqrt{481}}{-8} ina he tango te ±. Tango 6\sqrt{481} mai i -186.

$$x=\frac{-6\sqrt{481}-186}{-8}$$

Whakawehe -186-6\sqrt{481} ki te -8.

$$x=\frac{3\sqrt{481}+93}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{93-3\sqrt{481}}{4} x=\frac{3\sqrt{481}+93}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tāpirihia te 91 ki te 2, ka 93.

$$2x\left(93-2x\right)=1080$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te 93-2x.

$$186x-4x^{2}=1080$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-4x^{2}+186x=1080$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4x^{2}+186x}{-4}=\frac{1080}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$x^{2}+\frac{186}{-4}x=\frac{1080}{-4}$$

Whakahekea te hautanga \frac{186}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{93}{2}x=\frac{1080}{-4}$$

Whakawehe 1080 ki te -4.

$$x^{2}-\frac{93}{2}x=-270$$

Whakawehea te -\frac{93}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{93}{4}. Nā, tāpiria te pūrua o te -\frac{93}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{93}{2}x+\left(-\frac{93}{4}\right)^{2}=-270+\left(-\frac{93}{4}\right)^{2}$$

Pūruatia -\frac{93}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{93}{2}x+\frac{8649}{16}=-270+\frac{8649}{16}$$

Tāpiri -270 ki te \frac{8649}{16}.

$$x^{2}-\frac{93}{2}x+\frac{8649}{16}=\frac{4329}{16}$$

Tauwehea x^{2}-\frac{93}{2}x+\frac{8649}{16}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{93}{4}\right)^{2}=\frac{4329}{16}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{93}{4}\right)^{2}}=\sqrt{\frac{4329}{16}}$$

Whakarūnātia.

$$x-\frac{93}{4}=\frac{3\sqrt{481}}{4} x-\frac{93}{4}=-\frac{3\sqrt{481}}{4}$$

Me tāpiri \frac{93}{4} ki ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{481}+93}{4} x=\frac{93-3\sqrt{481}}{4}$$